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Calculus BC: Applications of the Derivative

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Suppose that an object is confined to move in a straight line, and that the function
f (t)
represents the position of the object relative to a fixed coordinate system at a time
t
. For
example, a marble might be released and allowed roll along a groove between floor boards,
f (t)
representing its signed distance in centimeters from the point of release, and
t
the
elapsed time in seconds.

The derivative
f'(t)
represents the rate of change of the position
f (t)
at time
t
, which
is the instantaneous velocity of the object. This is also a signed quantity, with the sign
indicating the direction of motion -- toward or away from the chosen origin. The absolute
value of the velocity,
| f'(t)|
, is the speed of the object, which reflects how quickly it
is moving regardless of direction.

The second derivative of the position function,
f''(t)
, represents the rate of change of
velocity, which is acceleration. In our example, if the marble moves from a flat to sloped
region of the floor, it will begin to pick up speed, and
f''(t)
will become positive.

In kinematics, one learns that the acceleration of an object
is related to the forces acting on it. Therefore, if one observes the motion of an object
(in effect measuring
f (t)
), differentiation allows one to determine what forces were acting
upon it during the observed motion.